Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5175,2,Mod(1,5175)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5175, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5175.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5175 = 3^{2} \cdot 5^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5175.a (trivial) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | yes |
| Analytic conductor: | \(41.3225830460\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - x^{6} - 13x^{5} + 11x^{4} + 33x^{3} - 9x^{2} - 13x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 345) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{7} - x^{6} - 13x^{5} + 11x^{4} + 33x^{3} - 9x^{2} - 13x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{6} + 4\nu^{5} + 11\nu^{4} - 34\nu^{3} - 11\nu^{2} - 18\nu - 3 ) / 20 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -3\nu^{6} + 2\nu^{5} + 43\nu^{4} - 22\nu^{3} - 133\nu^{2} + 16\nu + 41 ) / 20 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 4\nu^{6} - 6\nu^{5} - 49\nu^{4} + 66\nu^{3} + 94\nu^{2} - 58\nu - 3 ) / 10 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 9\nu^{6} - 6\nu^{5} - 119\nu^{4} + 66\nu^{3} + 319\nu^{2} - 28\nu - 113 ) / 20 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 6\nu^{6} - 9\nu^{5} - 76\nu^{4} + 104\nu^{3} + 176\nu^{2} - 147\nu - 62 ) / 10 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -7\nu^{6} + 8\nu^{5} + 87\nu^{4} - 88\nu^{3} - 187\nu^{2} + 84\nu + 39 ) / 10 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{6} + \beta_{5} + 2\beta_{4} - \beta_{3} + 3\beta_{2} - \beta _1 + 7 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 8\beta_{6} + \beta_{5} + 7\beta_{4} + 8\beta_{3} + 8\beta_{2} + 3\beta _1 + 1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{6} + 4\beta_{5} + 9\beta_{4} - 5\beta_{3} + 17\beta_{2} - 4\beta _1 + 27 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 67\beta_{6} + 6\beta_{5} + 61\beta_{4} + 71\beta_{3} + 71\beta_{2} + 38\beta _1 + 2 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 33\beta_{6} + 67\beta_{5} + 164\beta_{4} - 105\beta_{3} + 335\beta_{2} - 67\beta _1 + 485 ) / 2 \)
|
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.63914 | 0 | 4.96506 | 0 | 0 | 3.29106 | −7.82519 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.40464 | 0 | −0.0269736 | 0 | 0 | 1.58054 | 2.84718 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.27208 | 0 | −0.381820 | 0 | 0 | −3.58355 | 3.02986 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.161527 | 0 | −1.97391 | 0 | 0 | 5.15575 | −0.641892 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.93829 | 0 | 1.75698 | 0 | 0 | −3.86288 | −0.471035 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.43902 | 0 | 3.94880 | 0 | 0 | −0.839790 | 4.75315 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.77702 | 0 | 5.71186 | 0 | 0 | 3.25887 | 10.3079 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(23\) | \( -1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 5175.2.a.ch | 7 | |
| 3.b | odd | 2 | 1 | 1725.2.a.bk | 7 | ||
| 5.b | even | 2 | 1 | 5175.2.a.ca | 7 | ||
| 5.c | odd | 4 | 2 | 1035.2.b.f | 14 | ||
| 15.d | odd | 2 | 1 | 1725.2.a.bl | 7 | ||
| 15.e | even | 4 | 2 | 345.2.b.d | ✓ | 14 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 345.2.b.d | ✓ | 14 | 15.e | even | 4 | 2 | |
| 1035.2.b.f | 14 | 5.c | odd | 4 | 2 | ||
| 1725.2.a.bk | 7 | 3.b | odd | 2 | 1 | ||
| 1725.2.a.bl | 7 | 15.d | odd | 2 | 1 | ||
| 5175.2.a.ca | 7 | 5.b | even | 2 | 1 | ||
| 5175.2.a.ch | 7 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5175))\):
|
\( T_{2}^{7} - 2T_{2}^{6} - 12T_{2}^{5} + 20T_{2}^{4} + 43T_{2}^{3} - 44T_{2}^{2} - 56T_{2} + 10 \)
|
|
\( T_{7}^{7} - 5T_{7}^{6} - 27T_{7}^{5} + 141T_{7}^{4} + 158T_{7}^{3} - 1068T_{7}^{2} + 296T_{7} + 1016 \)
|
|
\( T_{11}^{7} + 12T_{11}^{6} + 8T_{11}^{5} - 338T_{11}^{4} - 864T_{11}^{3} + 1728T_{11}^{2} + 3840T_{11} - 5120 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 2 T^{6} + \cdots + 10 \)
|
| $3$ |
\( T^{7} \)
|
| $5$ |
\( T^{7} \)
|
| $7$ |
\( T^{7} - 5 T^{6} + \cdots + 1016 \)
|
| $11$ |
\( T^{7} + 12 T^{6} + \cdots - 5120 \)
|
| $13$ |
\( T^{7} - 2 T^{6} + \cdots + 128 \)
|
| $17$ |
\( T^{7} - 11 T^{6} + \cdots - 440 \)
|
| $19$ |
\( T^{7} - 16 T^{6} + \cdots + 2000 \)
|
| $23$ |
\( (T - 1)^{7} \)
|
| $29$ |
\( T^{7} - T^{6} + \cdots - 10000 \)
|
| $31$ |
\( T^{7} - 5 T^{6} + \cdots + 128 \)
|
| $37$ |
\( T^{7} - 17 T^{6} + \cdots - 16 \)
|
| $41$ |
\( T^{7} + 19 T^{6} + \cdots + 6640 \)
|
| $43$ |
\( T^{7} - 14 T^{6} + \cdots + 44320 \)
|
| $47$ |
\( T^{7} + 6 T^{6} + \cdots - 110720 \)
|
| $53$ |
\( T^{7} + 15 T^{6} + \cdots + 87560 \)
|
| $59$ |
\( T^{7} + 11 T^{6} + \cdots - 18320 \)
|
| $61$ |
\( T^{7} - 8 T^{6} + \cdots + 3215456 \)
|
| $67$ |
\( T^{7} + 3 T^{6} + \cdots - 653960 \)
|
| $71$ |
\( T^{7} - T^{6} + \cdots + 709040 \)
|
| $73$ |
\( T^{7} - 18 T^{6} + \cdots + 2048 \)
|
| $79$ |
\( T^{7} + 2 T^{6} + \cdots + 6400 \)
|
| $83$ |
\( T^{7} + T^{6} + \cdots - 2560 \)
|
| $89$ |
\( T^{7} + 16 T^{6} + \cdots + 272960 \)
|
| $97$ |
\( T^{7} - 26 T^{6} + \cdots + 955456 \)
|
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